On the Barzilai and Borwein choice of steplength for the gradient method

In a recent paper, Barzilai and Borwein presented a new choice of steplength forthe gradient method. Their choice does not guarantee descent in the objectivefunction and greatly speeds up the convergence of the method. They presented aconvergence analysis of their method only in the two-dimensional quadratic case.We establish the convergence of the Barzilai and Borwein gradient method whenapplied to the minimization of a strictly convex quadratic function of any numberof variables.1. Introduction